Average signature and 4-genus of 2-bridge knots

TL;DR

This paper investigates the average signature and 4-genus of 2-bridge knots, providing asymptotic and upper bound results that deepen understanding of their geometric and algebraic properties.

Contribution

It establishes the asymptotic behavior of average signatures and derives a new sublinear upper bound for the average 4-genus of 2-bridge knots.

Findings

Average signature approaches rac{rac{2c}{\u00a0 ext{pi}}}

Average 4-genus is sublinear in crossing number

Derived specific upper bound of 9.75c/a0log c

Abstract

We show that the average or expected absolute value of the signatures of all 2-bridge knots with crossing number $c$ approaches $\sqrt{{2c}/{\pi}}$. Baader, Kjuchukova, Lewark, Misev, and Ray consider a model for 2-bridge knot diagrams indexed by diagrammatic crossing number $n$ and show that the average 4-genus is sublinear in $n$. We build upon this result in two ways to obtain an upper bound for the average 4-genus of a 2-bridge knot: our model is indexed by crossing number $c$ and gives a specific sublinear upper bound of $9.75c/\log c$.